Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T11:14:37.508Z Has data issue: false hasContentIssue false
  • English
  • Français

A C0-Collocation-like method for elliptic equations on rectangular regions

Published online by Cambridge University Press:  17 February 2009

Zbigniew Leyk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 3 , January 1997 , pp. 368 - 387
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bramble, J. H. and Hilbert, S. R., “Estimation of linear functional on Sobolev spaces with application to Fourier transforms and spline interpolation”, SIAM J. Numer. Anal. 7 (1970) 113–124.CrossRefGoogle Scholar
[2]Ciarlet, P., The finite element method for elliptic problems (North-Holland, Amsterdam, 1978).Google Scholar
[3]Davis, P. J. and Rabinowitz, P., Methods of numerical integration (Academic Press, New York, 1975).Google Scholar
[4]de Boor, C. and Swartz, B., “Collocation at Gaussian points”, SIAM J. Numer. Anal. 10 (1973) 582–606.CrossRefGoogle Scholar
[5]Díaz, J. C., “A collocation-Galerkin method for the two point boundary value problem using continuous piecewise polynomial spaces”, SIAM J. Numer. Anal. 14 (1977) 844–858.Google Scholar
[6]Díaz, J. C., “A collocation-Galerkin method for Poisson equation on rectangular regions”, Math. Comput. 33 (1979) 77–84.Google Scholar
[7]Dunn, R. and Wheeler, M. F., “Some collocation-Galerkin methods for two-point boundary value problems”, SIAM J. Numer. Anal. 13 (1976) 720–733.CrossRefGoogle Scholar
[8]Dyksen, W. R., “Tensor product generalized ADI methods for separable elliptic problems”, SIAM J. Numer. Anal. 1 (1987) 59–76.Google Scholar
[9]Ladyzhenskaia, O. A. and Ural'ceva, N. N., Linear and quasi-linear equations of elliptic type, (in Russian) (Nauka, Moscow, 1964).Google Scholar
[10]Leyk, Z., “C 0-collocation-like method for two-point boundary value problems”, Numer. Math. 49 (1986) 39–53.Google Scholar
[11]Leyk, Z., “C 0-collocation-like methods at Radau points for two-point boundary value problems”, in Numerical Methods (eds. Greenspan, D. and Rozsa, P.), (North-Holland, 1988), 287–305.Google Scholar
[12]Percell, P. and Wheeler, M. F., “A C 1 finite element collocation method for elliptic equations”, SIAM J. Numer. Anal. 17 (1980) 605–622.Google Scholar
[13]Prenter, P. M. and Russell, R. D., “Orthogonal collocation for elliptic partial differential equations”, SIAM J. Numer. Anal. 13 (1976) 923–939.Google Scholar
[14]Strang, G. and Fix, G. J., An analysis of the finite element method (Prentice-Hall, Englewood Cliffs, 1973).Google Scholar
[15]Wheeler, M. F., “A C 0-collocation-finite element method for two point boundary value problems and one space dimensional parabolic problems”, SIAM J. Numer. Anal. 14 (1977) 71–90.CrossRefGoogle Scholar